Locally Perturbed Random Walks with Unbounded Jumps

TL;DR

This paper extends the convergence results of random walks with local impurities to those with unbounded jumps, including infinite variance cases, showing they still converge to Brownian motion under appropriate scaling.

Contribution

It generalizes previous results to unbounded random walks with jumps in the domain of attraction of the normal law, including infinite variance cases.

Findings

Convergence to Brownian motion for finite variance unbounded walks.

Extension to non-normal domain of attraction with superdiffusive scaling.

Applicable to Lorentz-process-like random walks with infinite variance.

Abstract

In \cite{SzT}, D. Sz\'asz and A. Telcs have shown that for the diffusively scaled, simple symmetric random walk, weak convergence to the Brownian motion holds even in the case of local impurities if $d \ge 2$. The extension of their result to finite range random walks is straightforward. Here, however, we are interested in the situation when the random walk has unbounded range. Concretely we generalize the statement of \cite{SzT} to unbounded random walks whose jump distribution belongs to the domain of attraction of the normal law. We do this first: for diffusively scaled random walks on $\mathbf Z^d$ $(d \ge 2)$ having finite variance; and second: for random walks with distribution belonging to the non-normal domain of attraction of the normal law. This result can be applied to random walks with tail behavior analogous to that of the infinite horizon Lorentz-process; these, in particular, have infinite variance, and convergence to Brownian motion holds with the superdiffusive $\sqrt{n \log n}$ scaling.